RESEARCH POSTER PRESENTATION DESIGN © 2011 www.PosterPresentations.co m Generalizing and Justifying: Pre-Service K-8 Teachers’ Strategies and Representations Inductive Reasoning (i.e. reasoning from specific cases to the general) is an important way of mathematical thinking. Multiple ways of thinking about mathematical concepts provide evidence of deeper understanding Introduction Objectives 17 Pre-Service Teachers 184 written solutions to pattern finding-tasks Qualitative analysis of solutions using specific rubric Evidence of Understanding Organization Methodology Results and Discussion Conclusions Pre-service teachers’ can solve pattern finding problems using various strategies They use at least one method of organization during problem-solving process Although generalizations are provided, the majority are based on numbers only; no links are made comparing numbers to structure Many justifications are made but most are invalid Bibliography Flores, A., Bright , G. & Joyner, J. (2009). Mathematics for every student: Responding to diversity, grades 9-12. Reston, VA: National Council of Teachers of Mathematics. Friel, S. & Markworth, K. (2009). "A Framework for Analyzing Geometric Pattern Tasks." Mathematics Teaching in the Middle School. 15.1. National Council of Teachers of Mathematics (1991). Professional Standards for Teaching Mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics (2000). Principle Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics (2011). Principle Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. Polya, G (1957). How to Solve It. Garden City, NY: Doubleday and Co., Inc. Acknowledgements Marquette University MSCS and NSF REU Drs. K. Factor and D. Brylow Directors, MSCS Dr. M. Magiera, Research Mentor To identify and describe pre-service K-8 teachers’ processes of generalizing and justifying: • Strategies • Representations • Use of visual/structural and numerical information Generalizations made with numbers only were the most frequently observed Invalid justifications were most frequently observed The use of one method of organization was most frequently observed Algebra & Arithmetic together were the most frequently observed Generalization Justifications Sample: Type of Work Reviewed Mentor Marta T. Magiera, PhD Kara S. Grier, REU 2011, [email protected] Systematic diagram and table to organize Use of algebra and arithmetic to solve • Simplified into explicit formula • Linked numbers to structure • Extended to ALL cases Gave a valid justification in explanation Data Analysis Algebraic Arithmetic Trial & Error Representation (table, graph, diagram or list) Systematic/Not Systematic Multiple Single None Data Analysis cont.… Structural Numerical None Valid Invalid None 26% 19% 0% 54% 1% 0% 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 Algebraic Arithmetic Trial & Error Algebra & Arithmetic Arithmetic & Trial Algebra & Trial Percent Responses Evidence of Understanding 27% 47% 24% 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 2 or more ways 1 only none Percent Responses Evidence of Organization 43% 52% 4% 0.00 20.00 40.00 60.00 80.00 100.00 Structural Numerical None Percent Responses Generalizing 39% 53% 7% 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 Valid Invalid None Percent Responses Justifications